#71
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Re: Heads Up Game Theory exercise
[ QUOTE ]
[ QUOTE ] [ QUOTE ] I nearly agree with this answer. This is a jam/fold problem with smallblind=1 bigblind=2. With stacksize=6 (SB can raise to 3x BB) I get a solution of SB: 1..16 jam 17 mixed 18..59 fold 60..100 jam BB: 1..62 fold 63 mixed 64..100 call Marv [/ QUOTE ] Marv, Assuming the mixed is 50%/50% jam/fold I don't think is correct. [/ QUOTE ] No, by mixed I did not mean exactly 50%/50% . Marv [/ QUOTE ] The master of being vague. Rather than tell me what you didn't mean, couldn't you have included what you did mean. Mixed doesn't always mean 50%/50%, but what does it mean to you in this situation? I can't see any mixed percentages for 17 that work with this strategy. SB: 1..16 jam 17 mixed 18..59 fold 60..100 jam The reason I assumed you meant 50%/50% is because if your strategy is only mixed for one #, then it has to be 50%/50% for this "game". |
#72
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Re: Heads Up Game Theory exercise
[ QUOTE ]
[ QUOTE ] [ QUOTE ] [ QUOTE ] I nearly agree with this answer. This is a jam/fold problem with smallblind=1 bigblind=2. With stacksize=6 (SB can raise to 3x BB) I get a solution of SB: 1..16 jam 17 mixed 18..59 fold 60..100 jam BB: 1..62 fold 63 mixed 64..100 call Marv [/ QUOTE ] Marv, Assuming the mixed is 50%/50% jam/fold I don't think is correct. [/ QUOTE ] No, by mixed I did not mean exactly 50%/50% . Marv [/ QUOTE ] The master of being vague. Rather than tell me what you didn't mean, couldn't you have included what you did mean. Mixed doesn't always mean 50%/50%, but what does it mean to you in this situation? I can't see any mixed percentages for 17 that work with this strategy. SB: 1..16 jam 17 mixed 18..59 fold 60..100 jam The reason I assumed you meant 50%/50% is because if your strategy is only mixed for one #, then it has to be 50%/50% for this "game". [/ QUOTE ] OK, with %ages (and hopefully no transcription errors on my part this time) SB: 1..16 jam 17 mixed: p(jam) = 0.5 18..60 fold 61..100 jam BB: 1..62 fold 63 mixed: p(call) = 0.125 64..100 call I get the value of the game as -0.157121 . Marv |
#73
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Re: Heads Up Game Theory exercise
For the SB, the sum of all Jam fractions from 1 to 62 must total 37/2
In the case of 1..16 jam 17 mixed 15..59 fold 60..100 jam The jam fractions for 1..16 and 60..62 (1/1 each) total 19/1. Since 17's jam fraction can't be a negative, the total can't be 37/2. I think the Best strategy of the many for SB would be 1..62 Jam and fold (37/124, 87/124) 63..100 Jam That would be the hardest for the BB to read since SB could have any card when it raises. |
#74
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Re: Heads Up Game Theory exercise
[ QUOTE ]
[ QUOTE ] [ QUOTE ] [ QUOTE ] [ QUOTE ] I nearly agree with this answer. This is a jam/fold problem with smallblind=1 bigblind=2. With stacksize=6 (SB can raise to 3x BB) I get a solution of SB: 1..16 jam 17 mixed 18..59 fold 60..100 jam BB: 1..62 fold 63 mixed 64..100 call Marv [/ QUOTE ] Marv, Assuming the mixed is 50%/50% jam/fold I don't think is correct. [/ QUOTE ] No, by mixed I did not mean exactly 50%/50% . Marv [/ QUOTE ] The master of being vague. Rather than tell me what you didn't mean, couldn't you have included what you did mean. Mixed doesn't always mean 50%/50%, but what does it mean to you in this situation? I can't see any mixed percentages for 17 that work with this strategy. SB: 1..16 jam 17 mixed 18..59 fold 60..100 jam The reason I assumed you meant 50%/50% is because if your strategy is only mixed for one #, then it has to be 50%/50% for this "game". [/ QUOTE ] OK, with %ages (and hopefully no transcription errors on my part this time) SB: 1..16 jam 17 mixed: p(jam) = 0.5 18..60 fold 61..100 jam BB: 1..62 fold 63 mixed: p(call) = 0.125 64..100 call I get the value of the game as -0.157121 . Marv [/ QUOTE ] Ok you changed 60 from jam to fold. With that, we agree. Your EV is in terms of $ with an assumed $2 big blind. We agree on the EV as well, since my EV was +0.078560 big blinds per hand |
#75
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Re: Heads Up Game Theory exercise
[ QUOTE ]
For the SB, the sum of all Jam fractions from 1 to 62 must total 37/2 In the case of 1..16 jam 17 mixed 15..59 fold 60..100 jam The jam fractions for 1..16 and 60..62 (1/1 each) total 19/1. Since 17's jam fraction can't be a negative, the total can't be 37/2. I think the Best strategy of the many for SB would be 1..62 Jam and fold (37/124, 87/124) 63..100 Jam That would be the hardest for the BB to read since SB could have any card when it raises. [/ QUOTE ] It's an interesting question which Nash strategy is 'best' in general. You're suggesting one with greatest support. As you say this most 'deceptive'. But they often have the drawback of being dominated by other nash strategies. In the current case by those which put a larger jam fraction on stronger hands. Marv |
#76
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Re: Heads Up Game Theory exercise
to marv, mikey and other guys who study stuff way over my head:
can you please tell me why you have SB bluffing with his worst hands? Intuitively it seems like "bluffing" with your better hands would make more sense. Also, "For the SB, the sum of all Jam fractions from 1 to 62 must total 37/2" I have no idea how you came up with this. Can you explain to a newbie (no I don't want the algebra worked out for me step-by-step, just the general algorithm)? |
#77
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Re: Heads Up Game Theory exercise
Read this for more information on my solution . The questions are almost identical . Also , note that my algebra error was corrected by tnixon who has the right answer posted .
You may use my equation set up to find one possible solution . http://forumserver.twoplustwo.com/showfl...=1#Post11953545 |
#78
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Re: Heads Up Game Theory exercise
[ QUOTE ]
[ QUOTE ] For the SB, the sum of all Jam fractions from 1 to 62 must total 37/2 In the case of 1..16 jam 17 mixed 15..59 fold 60..100 jam The jam fractions for 1..16 and 60..62 (1/1 each) total 19/1. Since 17's jam fraction can't be a negative, the total can't be 37/2. I think the Best strategy of the many for SB would be 1..62 Jam and fold (37/124, 87/124) 63..100 Jam That would be the hardest for the BB to read since SB could have any card when it raises. [/ QUOTE ] It's an interesting question which Nash strategy is 'best' in general. You're suggesting one with greatest support. As you say this most 'deceptive'. But they often have the drawback of being dominated by other nash strategies. In the current case by those which put a larger jam fraction on stronger hands. Marv [/ QUOTE ] ok I see the difference. Strategy #1 1..62 Jam and fold (37/124, 87/124) 63..100 Jam Strategy #2 1..43 fold 44 jam and fold (1/2, 1/2) 45..100 jam They are both optimal in that thier worst case EV is the same. But if the BB doesn't play optimal, Strategy #2 punishes the BB's mistakes harder. |
#79
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Re: Heads Up Game Theory exercise
[ QUOTE ]
to marv, mikey and other guys who study stuff way over my head: can you please tell me why you have SB bluffing with his worst hands? Intuitively it seems like "bluffing" with your better hands would make more sense. Also, "For the SB, the sum of all Jam fractions from 1 to 62 must total 37/2" I have no idea how you came up with this. Can you explain to a newbie (no I don't want the algebra worked out for me step-by-step, just the general algorithm)? [/ QUOTE ] To the first point, you're right that bluffing with hands that are stronger than those you fold is better (since you'll have a better hand when oppo calls), but if we assume our opponent knows our strategy and responds optimally, then the only way that any of these bluffs can win is if he folds, since our hand is worse than the worst hand he will call with. Thus the solutions with very weak bluffs are optimal in the game theory sense (as we reduce the BB's advantage to its minimal amount) but it's not the one you'd want to use. Different algorithms for finding Nash equilibria often locate different optima. The one I used happened to pick about the worst! Marv |
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